Dark matter stability and Dirac neutrinos using only Standard Model symmetries Cesar Bonilla,1, ∗ Salvador Centelles-Chuli´ a,2, † Ricardo Cepedello,2, ‡ Eduardo Peinado,3, § and Rahul Srivastava2, ¶ 1
arXiv:1812.01599v1 [hep-ph] 4 Dec 2018
Physik-Department T30d, Technische Universit¨ at M¨ unchen. James-Franck-Strasse, 85748 Garching, Germany 2 AHEP Group, Institut de F´ısica Corpuscular – CSIC-Universitat de Val`encia, Parc Cient´ıfic de Paterna. C/ Catedr´ atico Jos´e Beltr´ an, 2 E-46980 Paterna (Valencia) - SPAIN 3 Instituto de F´ısica, Universidad Nacional Aut´ onoma de M´exico, A.P. 20-364, Ciudad de M´exico 01000, M´exico.
We provide a generic framework to obtain stable dark matter along with naturally small Dirac neutrino masses generated at the loop level. This is achieved through the spontaneous breaking of the global U (1)B−L symmetry already present in Standard Model. The U (1)B−L symmetry is broken down to a residual even Zn ; n ≥ 4 subgroup. The residual Zn symmetry simultaneously guarantees dark matter stability and protects the Dirac nature of neutrinos. The U (1)B−L symmetry in our setup is anomaly free and can also be gauged in a straightforward way. Finally, we present an explicit example using our framework to show the idea in action.
As of now, a plethora of cosmic observations all indicate that the bulk of matter in the universe is in the form of dark matter, a hitherto unknown form of matter which interacts gravitationally, but has little or no electromagnetic interaction [1]. Similarly, the observation of neutrino oscillations has conclusively proven the existence of mass for at least two active neutrinos [2–5]. These observations are two of the most serious shortcomings of Standard Model (SM) since in SM there is no viable candidate for dark matter and neutrinos are predicted to be massless. Thus, they both inarguably point to presence of new physics beyond SM and are topics of active theoretical and experimental research. To explain dark matter, the particle content of the SM needs to be extended. Furthermore, to account for dark matter stability, new explicit [6, 7] or accidental symmetries [8] beyond those of the SM are also invoked. On the other hand, the understanding of the tiny, yet nonzero, masses of neutrinos also requires extending the SM in one way or another [9, 10]. However, the type of SM extensions required to explain the neutrino masses depend crucially on the Dirac/Majorana nature of neutrinos. This still remains an open question despite tremen-
∗ † ‡ § ¶
[email protected] [email protected] [email protected] [email protected] [email protected]
dous amounts of experimental effort [11–14]. There are several ongoing and planned experiments searching for neutrinoless double beta decay [12, 13], which if observed, owing to the Black-Box theorem [15], would imply that neutrinos are Majorana particles. Conversion of µ− → e+ in muonic atoms is another way to probe nature of neutrinos [14]. Furthermore, inference about nature of neutrinos can also be derived if lepton number violating decays are observed at colliders [16]. In absence of any experimental or observational signature, the nature of neutrinos remains an open question. From a theoretical point of view, the issue of Dirac/Majorana nature of neutrinos is intimately connected with the U (1)B−L symmetry of the SM and its possible breaking pattern [17]. If the U (1)B−L symmetry is conserved in nature, then the neutrinos will be Dirac fermions. However, if it is broken to a residual Zm ; m ∈ Z+ , m ≥ 2 subgroup, Z+ being set of all positive integers, then the Dirac/Majorana nature will depend on the residual Zm symmetry. U (1)B−L → Zm ≡ Z2n+1 with n ∈ Z+ ⇒ Neutrinos are Dirac particles
U (1)B−L → Zm ≡ Z2n with n ∈ Z+
(1)
⇒ Neutrinos can be Dirac or Majorana If the U (1)B−L is broken to a Z2n subgroup, then one can make a further classification depending on how the
2 SM lepton doublets Li = (νLi , lLi )T transform, ( ω n under Z2n ⇒ Dirac Neutrinos Li ∼ ω n under Z2n ⇒ Majorana Neutrinos
(2)
Thus, Dirac neutrinos are more natural from symmetry point of view, contrary to the popular belief. Moreover, in some recent works it has been argued that in a full theory with weak gravity conjecture neutrinos are expected to be Dirac fermions [18]. Owing to the above arguments, in past few years, Dirac neutrinos have gained a significant amount of attention leading to development of several elegant mass generation mechanisms to obtain naturally small Dirac neutrino masses [19–28]. Coming back to dark matter, especially attractive are scenarios which connect dark matter to neutrino physics in an intimate manner. The scotogenic model is one such model where the “dark sector” participates in the loop responsible for neutrino mass generation [7]. Recently, a relation between Dirac nature of neutrinos and dark matter stability has also been established [29]. Furthermore, it has been shown that this relation is independent of the neutrino mass generation mechanism [10]. It utilizes the SM Lepton number U (1)L symmetry1 , or its appropriate Zn subgroup, to forbid Majorana mass terms of neutrinos as well as to stabilize dark matter [29]. In this approach, the Dirac nature of neutrinos and stability of dark matter are intimately connected, having their origin from the same lepton number symmetry, already present in SM. In this letter we aim to combine and generalize these two approaches and develop a general formalism where
I Neutrinos are Dirac in nature. II Naturally small neutrino masses are generated through finite loops, forbidding the tree-level neutrino Yukawa couplings. III The dark sector participates in the loop. The lightest particle being stable is a good dark matter candidate. Usually one needs at least three different symmetries besides those within the Standard Model to achieve this
1
One can equivalently use the anomaly free U (1)B−L symmetry
[22]. However, we show that all these requirements can be satisfied without adding any extra explicit or accidental symmetries. In our formalism we employ an anomaly free chiral realization of the U (1)B−L spontaneously broken to a residual Zn symmetry and show that just the U (1)B−L already present in SM is sufficient. Before going into the details of the formalism lets briefly discuss the possibility of chiral solutions to U (1)B−L anomaly cancellation conditions. It is well known that the accidental U (1)B and U (1)L symmetries of SM are anomalous, but the U (1)B−L combination can be made anomaly free by adding three right handed neutrinos νRi with (−1, −1, −1) vector charges under U (1)B−L symmetry. However, chiral solutions to U (1)B−L anomaly cancellation conditions are also possible. The particularly attractive feature of chiral solutions is that using them one can automatically satisfy conditions (I) and (II), as shown in [19, 20], using the chiral solution νRi ∼ (−4, −4, 5) under U (1)B−L symmetry. Our general strategy is to use the chiral anomaly free solutions of U (1)B−L symmetry to generate loop masses for Dirac neutrinos and also have stable dark matter particle mediating the aforementioned loop. Then, after symmetry breaking once all the scalars get a vev, the U (1)B−L symmetry will be broken down to one of its Zn subgroups, such that the dark matter stability and Dirac nature of neutrinos remains protected. This scheme is shown diagrammatically in Fig. 1. In Fig. 1 the SM singlet fermions NLi , NRi , as well as the right handed neutrinos νR , have chiral charges under U (1)B−L symmetry2 . In order to generate the masses of these chiral fermions we have also added SM singlet scalars χi which also carry U (1)B−L charges. To complete the neutrino mass generation loop, additional scalars ϕ, ηi carrying U (1)B−L charges are required. After spontaneous symmetry breaking (SSB) of U (1)B−L symmetry, all the scalars χi will acquire vacuum expectation values (vev) breaking U (1)B−L → Zn residual symmetry. After SSB the fermions NLi , NRi get mass through the vev of scalars χi while the neutrinos acquire a naturally small n-loop mass as shown in Fig. 1.
2
It is not necessary that all fermions NLi , NRi be chiral under U (1)B−L symmetry.
3 hHi(1)
H(0) n-loops
n-loops
ϕ(z)
ηi (yi ) η1 (y1 )
NR1 (x1 )
NL1 (x01 )
χ1 (ζ1 )
NRi (xi )
ηi (ω βi )
ϕ(ω ) η1 (ω β1 )
ηi−1 (yi−1 ) ...
L (−1)
σ
NLi (x0i )
ηi−1 (ω βi−1 ) ...
νR
L (ω a )
(`)
χi (ζi )
NR1 NL1 (ω α1 ) (ω α1 )
NRi NLi (ω αi ) (ω αi )
hχ1 i(1)
hχi i(1)
νR (ω a )
(b) General residual Zn charge assignment.
(a) General U (1)B−L charge assignment.
Figure 1: General charge assignment for any topology and its spontaneous symmetry breaking pattern.
In order to satisfy all the requirements listed before, several conditions must be applied. First of all, the model should be anomaly-free:
Additionally, for dark matter stability: • After symmetry breaking the U (1)B−L symmetry is broken down to a Zn subgroup. Only even Zn ; n > 2 subgroups can protect dark matter stability. The odd Zn subgroups invariably lead to dark matter decay 3 . The symmetry breaking pattern can be extracted as follows. First all the U (1) charges must be rescaled in such a way that all the charges are integers and the least common multiple of all the rescaled charges is 1. Defining n as the least common multiple of the charges of the scalars χi , it is easy to see that the U (1) will break to a remnant Zn . This n must be taken to be even as explained before, i.e. n ≡ l.c.m(ζi ) ∈ 2Z.
• The chiral charges of the fermions must be taken in such a way that the anomalies are canceled. In order to obtain non-zero but naturally small Dirac neutrino masses the conditions are: ¯ Hν ˜ R should be • The tree level Yukawa coupling L forbidden. This implies that apart from the SM lepton doublets Li no other fermion can have U (1)B−L charge of ±1. Furthermore, to ensure that the desired loop diagram gives the dominant contribution to the neutrino masses, all lower loop diagrams should also be forbidden by an appropriate choice of fermion and vev carrying scalar charges.
• Dark sector particles should neither mix with nor decay to SM particles or to vev carrying scalars.
• The operator leading to neutrino mass generation, ¯ c χ1 . . . χi νR , should be invariant under the i.e. LH SM gauge symmetries as well as under U (1)B−L . Following the charge convention of Fig. 1 the charges of the vev carrying scalars χi should be P such that i ζi = −1 − ` . • All the fermions and scalars running in the neutrino mass loop must be massive. Since the fermions will be in general chiral, this mass can only be generated via the coupling with a vev carrying scalar. For example, in the diagram in Fig.1 we should have −xi + x0i + ζi = 0. • To protect the Diracness of neutrinos, all the Majorana mass terms for the neutrino fields at all loops must be forbidden in accordance with (2).
• There are two viable dark matter scenarios depending on the transformation of the Standard Model fermions under the residual symmetry. – When all SM fields transform as even powers of ω, where ω n = 1, under the residual Zn , the 3
For odd Zn subgroups, there will always be an effective dark matter decay operator allowed by the residual odd Zn symmetry. Even then it is possible that such an operator cannot be closed within a particular model, thus pinpointing the existence of an accidental symmetry that stabilizes dark matter. Another possibility is that the dark matter candidate decays at a sufficiently slow rate. Thus for residual odd Zn symmetries, one can still have either a stable dark matter stabilized by an accidental symmetry or a phenomenologically viable decaying dark matter. In this letter, we will not explore such possibilities.
U (1)B−L
Z6
Fermions
Li νRi N Ll NRl
(2, −1/2) (1, 0) (1, 0) (1, 0)
−1 (−4, −4, 5) −1/2 −1/2
ω4 4 (ω , ω 4 , ω 4 ) ω5 ω5
Scalars
4 Fields SU (2)L ⊗ U (1)Y
H χ η ξ
(2, 1/2) (1, 0) (2, 1/2) (1, 0)
0 3 1/2 7/2
1 1 ω ω
Table I: Charge assignment for all the fields. Z6 is the residual symmetry in this example, with ω 6 = 1.
lightest particle transforming as an odd power will be automatically stable, irrespective of its fermionic or scalar nature. We will show an explicit example of this simple yet powerful idea later. – In the case in which all SM fermions transform as odd powers of the residual subgroup, it can
H (0)
NR (−1/2)
Given the long list of requirements, most of the possible solutions that lead to anomaly cancellation fail to satisfy some or most of them. Still we have found some simple one loop and several two loop solutions that can satisfy all the conditions listed above. In this letter, we demonstrate the idea in action for a simple solution in which the U (1)B−L symmetry is broken down to a residual Z6 symmetry. Several other examples including cases of U (1)B−L breaking to other even Zn symmetries will be discussed elsewhere. Realistic example Let us consider an extension of the SM by adding an extra Higgs singlet χ with a U (1)B−L charge of 3, along with an scalar doublet η, a singlet ξ and two vector-like fermions NLl and NRl , with l = 1, 2, all carrying nontrivial U (1)B−L charges as shown in Table I and depicted in the left-hand side of Figure 2.
hHi (1)
χ (3)
η (1/2)
L (−1)
be shown that all the odd scalars and the even fermions will be stable due to a combination of the remnant Zn and Lorentz symmetry.
η (ω)
ξ (7/2)
×
NL (−1/2)
hχi (1)
L (ω 4 )
νR (−4)
NR (ω 5 )
ξ (ω)
×
NL (ω 5 )
νR (ω 4 )
(b) Remnant Z6 charge assignment.
(a) U (1)B−L charge assignment.
Figure 2: Charge assignment for the example model and its spontaneous symmetry breaking pattern.
The neutrino interactions are described by the following Lagrangian, 0 ¯ ¯ i η˜NR + yli ¯R NL + h.c., (3) Lν = yil L NLl νRi ξ + Mlm N m l l
where η˜ = iτ2 η ∗ , with the indices i = 1, 2, 3 and l, m = 1, 2. The relevant part of the scalar potential for generating the Dirac neutrino mass is given by V ⊃ λD H † ηχξ ∗ + h.c., where λD is an dimensionless quartic coupling.
(4)
After spontaneous symmetry breaking two neutrinos acquire a mass through the loop depicted in Fig. 2. Note that only νR1 and νR2 can participate in this mass gener0 ation due to the chiral charges (−4, −4, 5), i.e. yl3 = 0 in (3). The third right-handed neutrino νR3 remains massless and decouples from the rest of the model, although it is trivial to extend this simple model to generate its mass. The neutral component of the gauge doublet η and
5 the singlet ξ are rotated into the mass eigenbasis with eigenvalues m2i in the basis of (ξ, η 0 ). The neutrino mass is then given in terms of the one loop Passarino-Veltman function B0 [30] by, 2
Mν ∼
X λD vvχ 1 yy 0 2 M (−1)i B0 (0, m2i , M 2 ). (5) 2 2 16π mξ − mη i=1
It is worth to mention that since the U (1)B−L is anomaly free, it can be gauged. Then the physical Nambu-Goldstone boson associated to the dynamical generation of the Dirac neutrino mass [31] is absent. Regarding dark matter stability in this particular model, we can see that the lightest particle inside the loop is stable. This is true both for the fermionic and scalar dark matter candidates. As can be seen in Fig. 2b, all the internal loop particles are odd under the remnant Z6 , while all the SM particles are even. Therefore any combination of SM fields will be even under the remnant subgroup, forbidding all effective operators leading to dark matter decay. This simple but powerful idea can be seen graphically in Fig 3.
Figure 3: The decay of dark matter (odd under Z6 ) to SM particles (all even under Z6 ) is forbidden by the residual Z6 symmetry. This argument can be generalized to any even Zn symmetry.
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To summarize, we have shown that using the U (1)B−L symmetry already present in the Standard Model , it is possible to address the dark matter stability and relate it with the smallness of Dirac neutrino masses. We have described a general framework in which these features are realized exploiting the anomaly free chiral solutions of a global U (1)B−L . This framework can be utilized in a wide variety of scenarios. We have presented a particular simple realization of this idea, where neutrino masses are generated at one loop level and the U (1)B−L symmetry is broken spontaneously to a residual Z6 symmetry. The framework can also be used in models with higher order loops as well as to cases where U (1)B−L symmetry is broken to other even Zn subgroups. Since the U (1)B−L is anomaly free, it can be gauged in a straightforward way, giving a richer phenomenology from the dark matter and collider point of view.
ACKNOWLEDGMENTS
EP would like to thank the group AHEP (IFIC) and TUM for their hospitality during his visits. RS would like to thank IFUNAM for the warm hospitality during his visit. This work is supported by the Spanish grants SEV-2014-0398, FPA2017-85216-P (AEI/FEDER, UE), Red Consolider MultiDark FPA2017-90566-REDC and PROMETEOII/2014/084 (Generalitat Valenciana). The work of C.B. was supported by the Collaborative Research Center SFB1258. SCC is supported by the Spanish grant BES-2016-076643. RC is supported by the Spanish grant FPU15/03158. EP is supported in part by DGAPA-PAPIIT IN107118, the German-Mexican research collaboration grant SP 778/4-1 (DFG) and 278017 (CONACyT) and PIIF UNAM. RS will like to dedicate this paper to his grandparents, who despite not knowing what he is doing, never stopped believing in him.
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